Consider the following blood inventory problem facing a hospital. There is need for a rare blood type, namely, type AB, Rh negative blood. The demand D (in pints) over any 3-day period is given by
P{D = 0} = 0.4, P{D = 1} = 0.3,
P{D = 2} = 0.2, P{D = 3} = 0.1.
The expected demand is 1 pint, since E(D) = 0.3(1) + 0.2(2) + 0.1(3) = 1. Suppose that there are 3 days between deliveries. The hospital proposes a policy of receiving 1 pint at each delivery and using the oldest blood first. If more blood is required than is on hand, an expensive emergency delivery is made. Blood is discarded if it is still on the shelf after 21 days. Denote the state of the system as the number of pints on hand just after a delivery.
Thus, because of the discarding policy, the largest possible state is 7.
(a) Construct the (one-step) transition matrix for this Markov chain.
(b) Find the steady-state probabilities of the state of the Markov chain.
(c) Use the results from part (b) to find the steady-state probability that a pint of blood will need to be discarded during a 3-day period. Because the oldest blood is used first, a pint reaches 21 days only if the state was 7 and then D = 0.
(d) Use the results from part (b) to find the steady-state probability that an emergency delivery will be needed during the 3-day period between regular deliveries.